models of low-mass helium white dwarfs including gravitational … · 2016-06-17 · astronomy...

Astronomy & Astrophysics manuscript no. iptnsg_v3 c©ESO 2016June 17, 2016

Models of low-mass helium white dwarfs including gravitationalsettling, thermal and chemical diffusion, and rotational mixing

A. G. Istrate1, 2,?, P. Marchant1, T. M. Tauris3, 1, N. Langer1, R. J. Stancliffe1 and L. Grassitelli1

1 Argelander-Institute für Astronomie, Universität Bonn, Auf dem Hügel 71, 53121 Bonn, Germany2 Center for Gravitation, Cosmology and Astrophysics, Department of Physics, University of Wisconsin-Milwaukee, P.O. Box 413,

Milwaukee, WI 53201, USA3 Max-Planck-Institut für Radioastronomie, Auf dem Hügel 69, 53121 Bonn, Germany

Received May, 2016, Accepted June, 2016

ABSTRACT

A large number of extremely low-mass helium white dwarfs (ELM WDs) have been discovered in recent years. The majority ofthem are found in close binary systems suggesting they are formed either through a common-envelope phase or via stable masstransfer in a low-mass X-ray binary (LMXB) or a cataclysmic variable (CV) system. Here, we investigate the formation of theseobjects through the LMXB channel with emphasis on the proto-WD evolution in environments with different metallicities. We studyfor the first time the combined effects of rotational mixing and element diffusion (e.g. gravitational settling, thermal and chemicaldiffusion) on the evolution of proto-WDs and on the cooling properties of the resulting WDs. We present state-of-the-art binary stellarevolution models computed with MESA for metallicities of Z = 0.02, 0.01, 0.001 and 0.0002, producing WDs with masses between∼ 0.16 − 0.45 M. Our results confirm that element diffusion plays a significant role in the evolution of proto-WDs that experiencehydrogen shell flashes. The occurrence of these flashes produces a clear dichotomy in the cooling timescales of ELM WDs, which hasimportant consequences e.g. for the age determination of binary millisecond pulsars. In addition, we confirm that the threshold mass atwhich this dichotomy occurs depends on metallicity. Rotational mixing is found to counteract the effect of gravitational settling in thesurface layers of young, bloated ELM proto-WDs and therefore plays a key role in determining their surface chemical abundances,i.e. the observed presence of metals in their atmospheres. We predict that these proto-WDs have helium-rich envelopes through asignificant part of their lifetime. This is of great importance as helium is a crucial ingredient in the driving of the κ−mechanismsuggested for the newly observed ELM proto-WD pulsators. However, we find that the number of hydrogen shell flashes and, as aresult, the hydrogen envelope mass at the beginning of the cooling track, are not influenced significantly by rotational mixing. Inaddition to being dependent on proto-WD mass and metallicity, the hydrogen envelope mass of the newly formed proto-WDs dependson whether or not the donor star experiences a temporary contraction when the H-burning shell crosses the hydrogen discontinuityleft behind by the convective envelope. The hydrogen envelope at detachment, although small compared to the total mass of the WD,contains enough angular momentum such that the spin frequency of the resulting WD on the cooling track is well above the orbitalfrequency.

Key words. white dwarfs — stars: low-mass — stars: evolution — binaries: close — binaries: X-rays — pulsars: general

1. Introduction

Extremely low-mass white dwarfs (ELM WDs) are low-masshelium-core WDs with masses below 0.2−0.3 M and with sur-face gravities of 5 < log g < 7 (Brown et al. 2013). A large num-ber of such objects have been discovered in recent years throughdedicated or general surveys such as ELM, SPY, WASP, SDSSand the Kepler mission (e.g. Brown et al. 2010; Kilic et al. 2011;Brown et al. 2012; Kilic et al. 2012; Brown et al. 2013; Koesteret al. 2009; Maxted et al. 2011; Kepler et al. 2016; Brown et al.2016). Soon after the discovery of the first ELM WDs, it wasrecognised that they have to be a product of binary evolution(Marsh et al. 1995). From an evolutionary point of view, theseELM WDs cannot be formed from single-star progenitors asthe nuclear evolution timescale of such low-mass objects wouldexceed the Hubble time – unless they have an extremely highmetallicity (Kilic et al. 2007) or the star lost its envelope froman inspiralling giant planet (Nelemans & Tauris 1998). Indeed,the vast majority of ELM WDs are found in binary systems with

? e-mail: [email protected]

a companion star such as a neutron star in millisecond pulsar(MSP) systems (van Kerkwijk et al. 2005), an A-type star inEL CVn-type systems (Maxted et al. 2014a) or another (typicallya carbon-oxygen) WD. ELM WDs have been discovered in var-ious environments, from the Galactic disk to open and globularclusters (Rivera-Sandoval et al. 2015; Cadelano et al. 2015), andthus they can be formed from progenitors with different metal-licities.

The revived interest in ELM WDs was fostered by the dis-covery of pulsations in several of these objects (Hermes et al.2012b, 2013a,b; Kilic et al. 2015) as well as ELM proto-WDs(Maxted et al. 2013, 2014b; Corti et al. 2016; Gianninas et al.2016). The ELM WD pulsators extend the ZZ Ceti instabilitystrip to lower effective temperatures and higher luminosities.This instability strip contains stars with a convective drivingmechanism for pulsations acting at the base of the convectivezone associated with hydrogen recombination (e.g. Van Grootelet al. 2013). In the newly discovered ELM proto-WD pulsators,the excitation mechanism is instead the usual κ− mechanism forwhich the presence of He in the envelope is thought to play akey role (Jeffery & Saio 2013; Córsico et al. 2016). The pulsa-

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tional behaviour of ELM WDs and ELM proto-WDs provide anunique insight into their interior properties, such as the hydrogenenvelope mass and their total mass and rotation rate, which willplace stronger constraints on the theoretical models (e.g. Córsico& Althaus 2014a,b; Córsico et al. 2016).

Another interesting and not completely understood featureof ELM WDs is the observed presence of metals in their atmo-spheres. Gianninas et al. (2014a) provided for the first time sys-tematic measurements of the atmospheric abundances of He, Caand Mg for this type of stars and examined their distribution as afunction of effective temperature and mass. In the observed sam-ple, all the WDs with log g < 5.9 show Ca II K lines, suggestingthat the presence of metals in these objects is a ubiquitous phe-nomenon, possibly linked to their evolution. Detailed abundanceanalyses exist for only a handful of objects (Kaplan et al. 2013;Gianninas et al. 2014b; Hermes et al. 2014b; Latour et al. 2016)but already suggest a diversity of metallicities, as in the case ofsdB stars. Gravitational settling depletes the metals in the atmo-spheres of WDs on a very short timescale compared to their evo-lutionary timescale (Vauclair et al. 1979; Paquette et al. 1986;Koester et al. 2009), indicating that a process should be at workthat counteracts it or replenishes the depleted metals.

In addition to the formation and evolutionary history of theseobjects, their future outcome is also of theoretical interest. Short-period double WD binaries are candidate progenitors for tran-sient explosive phenomena such as Type Ia, underluminous .Iaand Ca-rich supernovae (Bildsten et al. 2007; Iben & Tutukov1984; Perets et al. 2010; Foley 2015), as well as exotic systemssuch as AM CVn stars, R Coronae Borealis (R CrB), and singlesubdwarf B/O stars (Kilic et al. 2014; Solheim 2010; Clayton2013; Heber 2016). Moreover, they are expected to be excellentsources of gravitational waves (Hermes et al. 2012a; Kilic et al.2013) and verification sources for gravitational detectors such aseLISA (Amaro-Seoane et al. 2012).

2. Formation and evolution of ELM WDs

From a theoretical point of view, an ELM WD can be formedeither through common-envelope evolution or stable Roche-lobe overflow (RLO) mass transfer in a low-mass X-ray binary(LMXB) or a cataclysmic variable (CV) system. The formationand evolution of low-mass WDs through a stable mass-transferphase (or by artificially removing envelope mass from its pro-genitor star) has been studied intensively over the years (e.g.Driebe et al. 1998; Sarna et al. 2000; Nelson et al. 2004; Al-thaus et al. 2001a; Panei et al. 2007; Althaus et al. 2013; Istrateet al. 2014a,b). In comparison, the common-envelope channel isless studied and far more uncertain (e.g. Nandez et al. 2015).

Although the majority of ELM WDs are found in double WDsystems (Andrews et al. 2014), almost all evolutionary calcula-tions that involve stable mass transfer producing an ELM WDconsider a neutron star companion (i.e. an LMXB progenitorsystem). For the structure of the final ELM WDs, the resultsof these LMXB calculations can also be applied to CV systemsproducing ELM WDs in double WD binaries, as the stellar prop-erties of the produced ELM WDs do not depend on the mass oftheir accreting companion, but instead on the initial orbital pe-riod and mass of the donor (progenitor) star (Nelson et al. 2004;De Vito & Benvenuto 2010; Istrate et al. 2014a). Only the orbitalperiods of the produced ELM WDs will be different.

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Fig. 1: Hertzsprung-Russel (HR) diagram showing the formationand cooling of a 0.28 M helium WD (produced in an LMXB)that undergoes a hydrogen shell flash. The initial progenitor massis 1.4 M (Z = 0.02), the neutron star mass is 1.2 M, and theinitial orbital period is 5.0 days. See Table 1 for ages at eachstage.

2.1. Hydrogen shell flashes and proto-WDs

After the RLO mass-transfer phase ends, the remaining donorstar goes through a so-called (bloated) proto-WD phase in whicha significant part of the hydrogen left in the envelope is burnedthrough stable hydrogen shell burning. In addition to this, de-pending on the mass of the proto-WD, its metallicity and thephysics included in the modelling, hydrogen may be burnedthrough short-lived phases of unstable burning through CNO hy-drogen shell flashes (e.g. Driebe et al. 1998; Althaus et al. 2001c;Nelson et al. 2004).

Figure 1 shows an example of the formation of an ELM WDthrough the LMXB channel, including the evolution as a proto-WD as well as its further cooling. The stellar track is computedfrom the zero-age main sequence (ZAMS) until the donor starreaches an age of 14 Gyr, points 0 and 12, respectively, in Fig. 1.In this case, the star experiences one hydrogen shell flash. Af-ter the Roche-lobe detachment (point 2), the proto-WD goesthrough a phase of contraction at almost constant luminosityand increasing effective temperature (between points 2 and 3).The total luminosity is dominated by CNO burning while thecontribution due to release of gravitational binding energy fromcontraction is negligible. When the proto-WD reaches point 3,which is at the beginning of the cooling branch, the tempera-ture in the burning shell is too low to sustain CNO burning,therefore the main contribution to the total luminosity is for awhile given by contraction, until the star switches to pp-burning.The unstable burning starts around point 4 and CNO burning be-comes dominant again. The increasing energy release during theflash development creates a steep temperature gradient close tothe location of maximum energy production. This will give riseto a pulse-driven convection zone within the hydrogen burningshell. After the convection zone is fully developed, the evolutionbecomes faster (between points 5 and 6). Around point 7, theconvection zone reaches the stellar surface, and consequently,its surface chemical composition is altered. The maximum hy-drogen luminosity reached during the flash is supplied by thepp-burning, although the onset of the instability is triggered bythe CNO cycling. Between points 7 and 8, the lower boundary ofthe pulse-driven convection zone moves upwards, and at point 8it completely vanishes. Beyond point 8, the contraction of the in-


Istrate et al.: A new grid of ELM WD models

Table 1: Evolution as a function of time for a 0.28 M proto-WDevolving through a hydrogen shell flash, as plotted in Fig. 1. TheZAMS is at point 0, and the onset of RLO (the LMXB phase) isat point 1. The relative age is in comparison to the previous pointof evolution and the WD age is with respect to the Roche-lobedetachment (point 2). See text for more details.

Point Relative age (Myr) WD age (Myr)0 – –1 3330 –2 217 03 4.92 4.924 6.63 11.55 57.1 68.76 0.0066 68.77 2.4× 10−5 68.78 3.8× 10−5 68.79 1.5× 10−5 68.7

10 3.1× 10−5 68.711 0.193 68.912 10 700 10 800

ner shells resumes, while the surface layers react by expansion,resulting in a redward motion in the HR-diagram that almostbrings the proto-WD back to the red-giant branch. At point 9,the star fills its Roche lobe again and a short episode of masstransfer is initiated (between points 9 and 10) with a high mass-transfer rate that approaches ∼10−7 M yr−1. After point 10, thestar again evolves towards a high surface temperature at almostconstant luminosity, and before reaching the final cooling track,it develops a so-called subflash (near log Teff = 4.4). The timeintervals for each of the above described phases are shown inTable 1.

The proto-WD phase has an associated timescale, ∆tproto,which is the time it takes the star to evolve (and contract) fromthe Roche-lobe detachment until it reaches its maximum effec-tive temperature on the (final) cooling track. In Fig. 1 this corre-sponds to the time interval during the evolution from point 2 topoint 11. The duration of this contraction phase is mainly givenby the burning rate of the residual hydrogen in the envelope.For evolved low-mass stars there is a well-known correlation be-tween the degenerate core mass and its luminosity (Refsdal &Weigert 1971). Therefore, after Roche-lobe detachment, the rateat which the residual hydrogen in the envelope is consumed is di-rectly proportional to the luminosity and thus increases stronglywith MWD. A detailed analysis of the dependence of ∆tproto on themass of the WD is given in Istrate et al. (2014b). This timescaleis especially important in MSP systems and should be added tothe optically determined cooling age of the WD to yield the trueage of the recycled radio pulsar. Unfortunately, the true age ofa recycled pulsar cannot be determined from its characteristicspin-down age as this method has proved unreliable by a factorof 10 or more (Camilo et al. 1994; Lorimer et al. 1995; Tauris2012; Tauris et al. 2012).

Hydrogen shell flashes occur in a range of proto-WD massesthat is dependent on the metallicity and whether or not elementdiffusion is included in the modelling. The lower mass limitfor flashes, Mflash,min, is determined by the size of the burningshell, such that if Mproto−WD < Mflash,min, then the shell is toothick to trigger unstable hydrogen burning. The upper mass limit,Mflash,max, is determined by the cooling time of the burning shell,which needs to be long enough to avoid an extinction of theshell before the instability is fully established (if Mproto−WD >

Mflash,max, this condition is not fulfilled, cf. Driebe et al. 1998).These conditions are altered when element diffusion is included(Althaus et al. 2001c), and this issue is investigated more care-fully in Sect. 4.

2.2. Age dichotomy in helium WD cooling?

The occurrence of hydrogen shell flashes, when element diffu-sion is taken into account, has been found to be responsible fora dichotomy in the cooling ages of helium WDs (Althaus et al.2001a; van Kerkwijk et al. 2005; Althaus et al. 2013; Bassa et al.2016). The occurrence of flashes in relatively massive heliumWDs (> 0.2 M), with initially thin hydrogen envelopes, leavesbehind an even thinner envelope, giving rise to relatively fastcooling. On the other hand, less massive (proto) helium WDs(< 0.2 M) have thicker hydrogen envelopes after RLO, re-sulting in stable shell hydrogen burning, and will therefore con-tinue residual hydrogen burning on the cooling track on a longtimescale.

Recently, Istrate et al. (2014b) found no evidence for sucha dichotomy in the case of thermal evolution of proto-WDs butrather a smooth transition with the mass of the WD. The au-thors showed that the thermal evolution timescale mainly de-pends on the proto-helium WD luminosity, which in turn de-pends on the mass of the proto-WD and not on the occurrence ofhydrogen shell flashes. These new findings questioned whethera dichotomy exists in the cooling ages of ELM WDs and if theresponsible process might be the occurrence of hydrogen shellflashes.

2.3. Aims of this investigation of ELM WDs

The focus of this paper is on the proto-WD phase of ELM WDs,which are investigated through a series of binary stellar evo-lution calculations of LMXBs. The following aspects are ad-dressed: (i) the hydrogen envelope mass as a result of binaryevolution, (ii) the role played by rotational mixing in the evolu-tion of ELM proto-WDs, (iii) the influence of element diffusionand rotation on ∆tproto as well as on the cooling timescale, (iv)the existence of a dichotomy in the cooling ages of ELM WDsas a result of the occurrence of hydrogen shell flashes, (v) thepresence of metals in the atmospheres of proto-WDs, and (vi)the relation between the mass of a proto-WD and its orbital pe-riod at the end of the mass-transfer phase. All these aspects areaddressed not only as a function of the proto-WD mass, but alsoas a function of metallicity. Answering these open questions isessential for understanding the formation of ELM WDs and theirage determination, for providing accurate models for astroseis-mology calculations, and for determining the correct age of MSPbinaries. This work extends the previous work by Istrate et al.(2014b) by including element diffusion and rotational mixing inthe evolution of the donor star and during the proto-WD and theWD cooling phase. Moreover, the study is extended to includethe effect of metallicity as well, for which we investigate fourmetallicities: Z = 0.02, 0.01, 0.001, and 0.0002.

3. Numerical methods

The evolutionary tracks presented in this paper are calculated us-ing the publicly available binary stellar evolution code MESA,version 7624 (Paxton et al. 2011, 2013, 2015). The nuclear net-work used is cno_extras.net and accounts for the CNO burn-ing with the following isotopes: 1H, 3He, 4He, 12C, 13C, 13N,



14N, 15N, 14O, 15O, 16O, 17O, 18O, 17F, 18F, 19F,18Ne, 19Ne, 20Ne,22Mg and 24Mg. Radiative opacities are taken from Fergusonet al. (2005) for 2.7 ≤ log T ≤ 4.5 and OPAL (Iglesias & Rogers1993, 1996) for 3.75 ≤ log T ≤ 8.7 and conductive opacitiesare adopted from Cassisi et al. (2007). Convective regions aretreated using the mixing-length theory (MLT) in the Henyeyet al. (1965) formulation with αMLT = 2.0. Transport of angu-lar momentum is treated as a diffusive process which results inrigid rotation in convective zones. The boundaries of convectiveregions are determined using the Schwarzschild criterion. A stepfunction overshooting extends the mixing region for 0.2 pres-sure scale heights beyond the convective boundary during coreH-burning.

We here refer to element diffusion as the physical mechanismfor mixing of chemical elements that is due to pressure gradients(or gravity, i.e. gravitational settling), temperature (thermal dif-fusion) and composition gradients (chemical diffusion). Gravita-tional settling tends to concentrate heavier elements towards thecentre of the star. Thermal diffusion generally acts in the samedirection, although to a lesser degree, by bringing highly chargedand more massive species towards the hottest region of the star(its centre). Chemical diffusion, on the other hand, has the op-posite effect (e.g. Iben & MacDonald 1985; Thoul et al. 1994).MESA includes the treatment of element diffusion through grav-itational settling, chemical and thermal diffusion (Thoul et al.1994), and radiative accelerations (Hu et al. 2011). Radiativeforces are proportional to the reciprocal of the temperature andare thus negligible in hot regions where nuclear burning is ofimportance. In addition, calculating these forces is computation-ally demanding. We therefore here neglected the effects of radia-tive levitation (which is important for determining photosphericcomposition of hot WDs (Fontaine & Michaud 1979). The de-tailed description of how element diffusion is implemented inMESA can be found in Paxton et al. (2015). We take into accountthe effects of element diffusion due to gravitational settling andchemical and thermal diffusion for the following elements 1H,3He, 4He, 12C, 13C, 14N, 16O, 20N, 24Mg, and 40Ca.MESA includes the effects of the centrifugal force on stellarstructure, chemical mixing, and transport of angular momen-tum that is due to rotationally induced hydrodynamic and sec-ular instabilities as described in Heger et al. (2000). Here, wetake into account the mixing due to dynamical shear instabil-ity, secular shear instability, Eddington-sweet circulation, andGoldreich-Schubert-Fricke instability with a mixing efficiencyfactor of fc = 1/30 (Heger et al. 2000). The mixing of angularmomentum that is due to dynamo-generated magnetic fields inradiative zones is also included (Spruit 2002; Heger et al. 2005)as is the angular momentum transport due to electron viscos-ity (Itoh et al. 1987). A decrease of the mean molecular weightwith radius has a damping effect on mixing processes driven byrotation or even prevents these from occurring. The strength ofthis effect is regulated by the parameter fµ, for which we followHeger et al. (2000) and set fµ=0.05.

The initial metallicity was set to Z = 0.02 (Y=0.28), withinitial abundances from Grevesse & Sauval (1998). The lowermetallicities were obtained by scaling both X and Y by the samefactor such that X + Y + Z = 1. For the WD evolution and forTeff < 10 000 K, the outer boundary conditions were derivedusing non-grey model atmospheres (Rohrmann et al. 2012).

To calculate the rate of change of orbital angular momentum,we took into account contributions from gravitational wave ra-diation, mass loss, magnetic braking, and spin orbit couplings:

Jorb = Jgwr + Jml + Jmb + Jls , (1)

as described in Paxton et al. (2015). The contribution of spin-orbit couplings to Jorb was computed by demanding conserva-tion of total angular momentum (except for losses due to grav-itational wave radiation, magnetic braking, and mass loss), thatis, changes in spin angular momentum were compensated for bychanging the orbital angular momentum. The initial rotation ve-locity of the donor star was set by requiring that its spin period besynchronized with the initial orbital period. The time evolutionof the angular velocity of the donor star is given by

dΩi

dt=

Ωorb −Ωi

τsync, (2)

where Ωi is the angular velocity of cell i (Detmers et al. 2008).The synchronization time, τsync was calculated using the formal-ism of tidal effects from Hurley et al. (2002) and depends onwhether the envelope is convective or radiative.

3.1. Grid of models

To produce our grid of models, we followed the detailed binaryevolution of the donor star from the ZAMS until it reached anage of 14 Gyr. The neutron star was treated as a point mass.The final outcome of these LMXB systems is very sensitive tothe initial orbital period and to the treatment of orbital angularmomentum loss (e.g. Istrate et al. 2014a, and references therein).

We calculated binary tracks for four metallicities: Z = 0.02,0.01, 0.001, and 0.0002. For each metallicity, the models weredivided into three categories: (i) basic models (with no diffusionnor rotation), (ii) diffusion models (with element diffusion) and,(iii) diffusion+rotation models (with element diffusion plus ro-tation). In both the diffusion and diffusion+rotation models, weincluded the effects of centrifugal forces and angular momentumtransport, which means that these two models only differ by thepresence of rotational mixing in the rotation models.

For Z = 0.02, the initial binary configuration has a 1.4 Mdonor star and a 1.2 M neutron star accretor. For all the othermetallicities, our models were calculated with a 1.0 M donorstar and a 1.4 M neutron star (to facilitate direct comparisonwith previous work in the literature, see Sect. 5.2). All the mod-els were computed using a magnetic braking index of γ = 4, andwe assumed that 30 per cent of the transferred mass is ejectedfrom the neutron star as a fast wind carrying its specific orbitalangular momentum. We note that the structure of the ELM WDsis not sensitive to the above choices of mass-transfer parame-ters which only affect their final orbital periods. A comprehen-sive study of the influence of the magnetic braking index andthe accretion efficiency on LMXB evolution can be found in Is-trate et al. (2014a). We point out again that we here refer to themass of the proto-WD as being the (bloated) donor star mass atthe end of the RLO mass-transfer phase (before the occurrenceof flashes, which can lead to additional mass-transfer episodes),and the mass of the WD as being the mass at the beginning of thecooling track. We calculated models just above the bifurcationperiod, which is defined as the shortest initial orbital period thatproduces a WD (e.g. Istrate et al. 2014a, and references therein).



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4. Results

4.1. General effects of element diffusion and rotationalmixing

In the context of low-mass helium WDs, element diffusion wasinvestigated in detail over the past few years by the La Platagroup (e.g. Althaus & Benvenuto 2000; Serenelli et al. 2001;Althaus et al. 2001a,c,b; Panei et al. 2007; Althaus et al. 2009,2013) using the stellar evolution code LPCODE for variousranges of helium WD masses and metallicity. To the best of ourknowledge, there is only one other study that used MESA forlow-mass helium WDs (Gautschy 2013). Our models include el-ement diffusion from the ZAMS and not only from the proto-WD phase, as in the previous works. Moreover, for the first time,we investigate in detail the role played by rotational mixing inaddition to element diffusion in the evolution of ELM WDs.

Element diffusion has a strong effect on the surface compo-sition of a proto-WD and on the chemical profile deep inside thestar close to the helium core. At the surface, gravitational set-

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Fig. 3: Evolution of hydrogen surface abundance (top panel) andlog g (bottom panel) for the three proto-WDs shown in Fig. 2illustrating the effect of gravitational settling, rotational mixingand the mixing due to convection zones developed during the hy-drogen shell flashes on the surface composition of these objects.

tling increases the hydrogen abundance given that hydrogen isthe lightest element. Close to the helium core boundary, chem-ical diffusion tends to smooth it out by mixing the hydrogendownwards into hotter layers because a large hydrogen abun-dance gradient exists. It has been shown that this hydrogen tailpromotes the occurrence of hydrogen shell flashes (e.g. Althauset al. 2001a). Moreover, when element diffusion is included, aproto-WD experiences more flashes than when diffusion is ne-glected (e.g. Althaus et al. 2001a). The WD mass interval inwhich they occur is also changed compared to the case whenelement diffusion is ignored. The number of flashes and otherinformation for all the models studied in this work are given inAppendix A.

Figure 2 shows the evolution of surface gravity versus effec-tive temperature for a proto-WD of ∼0.23 M obtained from thefollowing three model configurations: basic, diffusion, and diffu-sion+rotation. The basic model experiences three hydrogen shellflashes, while the diffusion and the diffusion+rotation models ex-perience one additional flash. The radial expansion following theCNO burning is more pronounced when element diffusion is in-cluded, in some cases leading to additional episodes of RLO. Ingeneral, the models with diffusion and diffusion+rotation behavein a very similar way.

Figure 3 shows the evolution of hydrogen surface abundance(top panel) and log g (bottom panel) for the same (proto)WDs asin Fig. 2. As already mentioned, gravitational settling changesthe surface abundances. All the elements heavier than hydro-gen sink below the surface, leaving a pure hydrogen envelopebehind. When rotational mixing is included, gravitational set-tling and rotational mixing compete with each other to deter-mine the chemical composition of the surface. At the beginningof the proto-WD phase, rotational mixing dominates. However,the surface gravity of the proto-WD increases with time, whilethe efficiency of rotational mixing decreases, as described inSect. 4.1.1. Thus, in later phases of the evolution, the gravita-tional settling overcomes the mixing induced by rotation. By the



(a)

(b) (c)

Fig. 4: Kippenhahn diagrams showing the proto-WD phase for the same systems as in Figure 2. The plots show cross sections of theouter ∼ 0.01 M envelope of the proto-WD in mass coordinates, along the y-axis, as a function of stellar age on the x-axis (relative tothe ZAMS age). The green areas denote zones with convection; the dotted white lines define lines of constant hydrogen abundance,10−2 to 10−5 (from top to bottom). The intensity of the blue and red colour indicates the hydrogen abundance by mass fraction, asshown on the colour scale to the right. As a result of different input physics, the proto-WDs have slightly different masses and ages.See text for details.

beginning of the last flash, the surface structure of the model thatonly includes element diffusion is nearly identical to the struc-ture of the model that includes both element diffusion and ro-tational mixing. Helium in the envelopes of ELM proto-WDsis a crucial ingredient for exciting pulsation modes through theκ−mechanism, as shown by Jeffery & Saio (2013) and Córsicoet al. (2016) for radial and nonradial modes. Gianninas et al.(2016) recently provided the first empirical evidence that pul-sations in ELM proto-WDs can only occur when a significantamount of helium is present in their atmospheres. In contrastwith evolutionary models that only include element diffusion,our new evolutionary models including rotational mixing pro-duce proto-WDs that have mixed He/H envelopes during mostof their evolution before settling on the cooling track.Another effect of element diffusion, resulting mainly from thecompetition between chemical and thermal diffusion, is the de-velopment of a hydrogen tail that reaches down into the hothelium-rich layers, as shown in Fig. 4. This effect is responsi-ble for the larger number of flashes compared to the case whereelement diffusion is ignored (basic model). Rotational mixing isseen not to change the chemical structure of the deep layers. Thiscan also be concluded from the very similar behaviour in termsof the number of flashes and the structure of the flashes in the

case that includes both diffusion and rotation compared to thecase that only includes element diffusion, cf. Figs. 2 and 5.

In Fig. 5 we plot the luminosity produced by hydrogen burn-ing versus the hydrogen envelope mass. For all three models,around 70 per cent of the hydrogen remaining from the end ofthe LMXB phase (Roche-lobe detachment) is processed beforethe occurrence of flashes while the bloated proto-WD crossesthe HR–diagram. The occurrence of additional flashes, whichapplies to the cases where element diffusion is included, reducesthe hydrogen envelope mass available on the cooling track (i.e.after reaching maximum Teff , marked by squares in Fig. 5) by afactor of ∼ 3 compared with the basic model. The basic modelstill experiences significant residual hydrogen burning on thecooling track. As a result, the basic model only cools down to atemperature of Teff ≈ 8400 K within 14 Gyr (since the ZAMS),while the two models that include diffusion will cool down toroughly Teff ≈ 4000 K (cf. Fig. 2). The cooling properties of theELM WDs are discussed in more detail in Sect. 4.5.

The orbital evolution of the models described above is shownin Fig. 6. One difference between the three models is that thosewith element diffusion (and rotation) require a longer initial or-bital period to form approximately the same proto-WD. The or-bital period at the Roche-lobe detachment is ∼7.05 days for the



Fig. 7: Kippenhahn diagram for the same proto-WD as in Fig. 4, including both element diffusion and rotational mixing. Theintensity of the orange and indigo colour indicates the ratio of the spin angular velocity to the orbital angular velocity, Ω/Ωorb, asshown on the colour scale to the right. The green areas and the dotted white lines have the same meaning as in Fig. 4. The blackarrows point to the position of the profiles in Fig. 8.

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Fig. 5: Hydrogen burning luminosity versus hydrogen envelopemass, for the same systems as in Fig. 2. The grey stars representthe moment of Roche-lobe detachment while the grey squaresdenote the point of maximum effective surface temperature. Theevolution is from the right to the left.

model that includes diffusion and rotation, ∼ 6.73 days for themodel with diffusion only, and ∼ 5.19 days for the basic model.As the diffusion-induced flashes are stronger and because almost

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Fig. 6: Orbital period evolution for the same models as in Fig. 2.The grey circles represent the onset of the mass transfer, thegrey stars represent Roche-lobe detachment and the grey squaresmark the maximum Teff .

every flash causes the star to expand and fill its Roche lobe again,the mass-transfer episodes widen the orbit during each flash. Inthe end, this effect accounts for an increase of a few per cent inthe orbital period.



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Fig. 8: Angular momentum diffusion coefficient, ν as function ofmass coordinate from the centre (left) to the surface of the star(right) for the same proto-WD as in Fig. 7. Processes includedare Spruit-Tayler dynamo (st), Eddington-Sweet circulation (es),and electron viscosity (visc). Each panel (from top to bottom)corresponds to a profile as marked by the black arrows in Fig. 7(from left to right). The grey dotted lines define lines of constanthydrogen abundance, from 10−5 to 10−2, while the black dashedline represents the hydrogen mass abundance.

4.1.1. Rotational mixing

As shown in Fig. 7, the tidal coupling is strong enough to com-pletely synchronize the donor with the orbit up until the end ofthe LMXB phase. This changes dramatically after detachmentfrom the Roche lobe; while the helium core barely contracts andspins up only slightly above the orbital frequency, the extendedhydrogen envelope spins up significantly during contraction, re-sulting in strong shear at the core-envelope boundary. As hydro-gen flashes develop and the star expands and then contracts, theenvelope successively spins down and up, with the rotational pe-riod at the surface of the proto-WD at its maximum being up to20 times shorter than the orbital period. Even though the proto-WD expands back and fills its Roche lobe during flashes, thesephases are very short. The convective layers developed duringthe flashes disappear well before the next phase of Roche–lobeoverflow such that tidal synchronization past the LMXB phaseis negligible.

Although a strong shear is developed, the composition gra-dients that help stabilize the instabilities driven by rotation pre-vent the mixing of elements and angular momentum from theenvelope to the core. This is shown in Fig. 8, where the differ-ent processes contributing to the angular momentum diffusioncoefficient, ν are shown at four different times. As depicted inthe first panel, there is a very steep H-gradient immediately af-ter a flash (and also after detachment from the LMXB phase) thatcompletely prevents mixing to the core. As burning proceeds be-tween flashes (second and third panels in Fig. 8), the H-gradientis softened and starts to move outwards in mass, which allowssome angular momentum to be transported to the core mainlythrough magnetic torques from the Spruit-Tayler dynamo. Fi-nally, after settling on the cooling track (final panels panel inFigs. 7 and 8), most of the remaining hydrogen has been burnt,and angular momentum has mixed efficiently between the enve-lope and the core. Despite the small mass of the envelope relativeto the total WD mass, this results in the WD having a spin periodmore than four times shorter (i.e. faster) than its orbital period.Because we have assumed that magnetic torques do not con-tribute to the mixing of elements, rotational mixing in our mod-els barely affects the formation of the hydrogen tail due to ele-ment diffusion, and thus has a weak effect on the strength andthe occurrence of flashes. At the surface, however, the fast rota-tion induces element mixing through Eddington-Sweet circula-tion (see Fig. 8), which counteracts the rapid settling of elementsheavier than hydrogen.

4.2. Effect of metallicity

For a given stellar mass, decreasing the metallicity produces adecrease in the radiative opacity which has an impact on thestellar evolution. The ZAMS and the RGB-phase are shifted to-wards the blue region of the HR-diagram, with the luminosityand effective temperature being higher during these phases thanfor models at solar metallicity. In other words, at low metallic-ity stars tend to be hotter, have smaller radii, and evolve morequickly than their high-metallicity counterparts. An immediateconsequence of lower metallicities is therefore a higher mass ofthe helium WD formed through the LMXB phase at a given ini-tial orbital period because the Roche lobe is filled at a more ad-vanced stage in the evolution as a result of the smaller radius.

As previously demonstrated by Serenelli et al. (2002) andNelson et al. (2004), the threshold mass for the occurrence ofhydrogen shell flashes increases with lower metallicity. This isconfirmed by our calculations, as shown in Table 2. For exam-



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Fig. 9: HR-diagram showing the formation and evolution of a∼0.28 M WD from progenitors with different metallicities. Thegrey symbols represent the beginning of RLO (circles), the endof RLO (stars) and the maximum Teff (squares). The grey dashedlines represent lines of constant radius.

ple, for the basic models (without diffusion and rotation), theminimum mass above which the flashes occur is ∼ 0.21 M forZ = 0.02, ∼0.22 M for Z = 0.01, ∼0.25 M for Z = 0.001, and∼0.28 M for Z = 0.0002.

Models with element diffusion have lower threshold valuesfor flashes to occur, with the following dependence on metallic-

Table 2: Proto-WD mass ranges for hydrogen shell flashes. For agiven model category and a given metallicity, the threshold massfor flashes also depends on the initial mass of the donor star.

Z category Mflash,min (M) Mflash,max (M)0.02 basic 0.212 0.305−0.3190.02 diffusion <0.167 >0.3920.02 diffusion+rotation <0.167 >0.3210.01 basic 0.222 0.305−0.3750.01 diffusion 0.167 >0.2910.01 diffusion+rotation <0.181 >0.320.001 basic 0.249 0.349−0.4220.001 diffusion 0.223 >0.3220.001 diffusion+rotation 0.232 >0.320.0002 basic 0.282 0.356−0.4410.0002 diffusion 0.266 >0.3110.0002 diffusion+rotation 0.275 >0.292

ity: all the models studied for Z = 0.02 experience flashes, thelowest mass proto-WD produced being 0.167 M. For Z = 0.01flashes occur above 0.169 M, for Z = 0.001 the limit is∼ 0.22 M, while for Z = 0.0002 the lower threshold value is∼ 0.26 M. When rotational mixing is included, all the thresh-old values are slightly higher than only diffusion is included, cf.Table 2. The upper limit for the occurrence of flashes is not aswell constrained because fewer models are calculated models inthis mass range, given that the focus of this work is towards thelowest masses of helium WDs, which are the ELM WDs. Theobtained limits for hydrogen shell flashes agree well with thosefound in the literature for low metallicity (Serenelli et al. 2001),but at solar metallicity we obtain somewhat lower values thanAlthaus et al. (2013).

The change in metallicity not only affects the threshold forflashes, but also the extent of the loops in the HR diagram. InFig. 9 the formation and evolution of a proto-WD with a massof ∼ 0.28 M is shown in the HR-diagram for all the inves-tigated metallicities. The lower the metal content, the weakerthe CNO burning, and thus the loops during the CNO flashesare markedly less extended than in models with higher metallic-ity. Moreover, the number of flashes increases with decreasingmetallicity: while the models for Z = 0.02 and Z = 0.01 ex-perience just one hydrogen shell flash, the model at Z = 0.001goes through two flashes, and at Z = 0.0002 the star experiencesthree hydrogen shell flashes. The interval of masses for whichflashes occur is also affected by metallicity. For Z = 0.02 andZ = 0.01, a 0.28 M helium WD is close to the upper masslimit where hydrogen shell flashes occur, while for Z = 0.001and Z = 0.0002, a 0.28 M helium WD is located close to thelower mass limit of the hydrogen shell flash interval. We stressthat the number of flashes decreases with increasing mass of theWD and varies between 0 and 7 flashes for our computed models(see Appendix A).

4.3. Inheritance of proto-WDs: the hydrogen envelope mass

Figure 10 shows the hydrogen envelope mass at the end of themass-transfer phase (Roche-lobe detachment), MH,det, as a func-tion of the proto-WD mass for all the computed models. Fora given metallicity, the models with diffusion and with diffu-sion+rotation have very similar values of MH,det as the basicmodels. The general trend is that the lower the mass of the proto-WD, the higher MH,det. The features in MH,det are given by theevolutionary history of the progenitor (donor) star and depend



on the point in its evolution at which mass transfer is initiated.We note a jump in the hydrogen envelope mass at ∼ 0.21 M,∼ 0.23 M, ∼ 0.29 M, and ∼ 0.34 M for Z = 0.02, Z = 0.01,Z = 0.001, and Z = 0.0002, respectively. This can be under-stood as discussed below. The shell hydrogen burning producesa convective envelope. When the convective envelope reachesits deepest extent, a hydrogen abundance gradient is producedbetween the region of the star mixed by the convective enve-lope and the layers below (which are rich in helium). When thehydrogen burning shell passes through this chemical disconti-nuity, the hydrogen burning rate drops, the radius contracts ona Kelvin-Helmholtz timescale and, as a result, the mass trans-fer will cease. The same phenomenon is responsible for the oc-currence of the luminosity bump in red-giant stars (e.g. Thomas1967; Christensen-Dalsgaard 2015), first discussed in the con-text of temporary Roche-lobe detachment in LMXBs in Tauris& Savonije (1999).

The interruption of the mass transfer can be a temporary ef-fect if the envelope is massive enough, such that when the burn-ing shell has passed through the discontinuity, the star still hasenough material to burn and can therefore resume its mass trans-fer. If its envelope has been stripped to a greater extent, then thedonor star is unable to resume mass transfer and a proto-WD isformed. This discontinuity in MH,det, observed at all the metallic-ities studied, distinguishes the systems that undergo this type oftemporary detachment (the systems on the right-hand or upperside of the discontinuity) from the systems in which the hydro-gen shell burning passes through the hydrogen abundance dis-continuity without being able to resume mass transfer afterwards(the systems on the left-hand or lower side of the discontinuity).This explains the increasing values of MH,det with Mproto−WD justbelow the discontinuity.

4.4. ∆tproto: the contraction timescale for proto-WDs

As has been discussed earlier in this work, after the end of theLMXB mass-transfer phase, a certain amount of time, ∆tproto, isrequired by the newly formed object, the proto-WD, to contractand reach its cooling track. This timescale, from Roche-lobe de-tachment to the beginning of the cooling track (defined as whenTeff reaches its maximum value), depends on the mass of theproto-WD and can reach up to 2 Gyr for the lowest mass proto-WDs (Istrate et al. 2014b) down to 10 − 100 Myr for the highestmass helium WDs. More importantly, Istrate et al. (2014b) haveshown that ∆tproto is not influenced by the occurrence of hydro-gen shell flashes, and consequently, the suggested dichotomy inWD cooling times produced by hydrogen flashes was called intoquestion.

The determination of ∆tproto is important, especially for de-termining the age of MSPs with helium WD companions in-dependently of the spin-down of the MSP (e.g. van Kerkwijket al. 2005; Antoniadis et al. 2012; Bassa et al. 2016). Duringthis phase, the proto-WD appears to be bloated, meaning that itsradius is significantly larger than the radius of a cold WD of sim-ilar mass. As the timescale for this contraction phase (∆tproto) ispredicted to be relatively long, a number of ELM WDs shouldbe observed in this bloated stage. One example suggested by Is-trate et al. (2014b) is PSR J1816+4510, based on observationsby Kaplan et al. (2012, 2013).

Figure 11 shows ∆tproto for all our computed models. Onefeature is the occurrence of clustering in the data that groupsthe proto-WDs that undergo the same number of flashes (seeAppendix A). As discussed before, the models with diffusiononly and diffusion+rotation behave in a very similar way. In gen-

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Fig. 10: Hydrogen envelope mass at the end of the mass-transferphase (Roche-lobe detachment) for the basic stars (purple cir-cles), for the stars with diffusion only (orange stars) and for thestars with diffusion+rotation (blue squares) for Z = 0.02 (toppanel), Z = 0.01 (second panel), Z = 0.001 (third panel) andZ = 0.0002 (bottom panel) as a function of proto-WD mass.The grey shaded area denotes the stars that undergo a temporaryRoche-lobe detachment (see text for details).

eral, ∆tproto is larger than in the basic models when diffusion isincluded because of the additional flashes. For Z = 0.02 andZ = 0.01 there is a smooth transition of ∆tproto around the limitof the occurrence of flashes (models that experienced hydrogenshell flashes are plotted with open symbols). We recall that forthese high metallicities all the models with diffusion only anddiffusion+rotation (except for the model with the lowest massat Z = 0.01) undergo unstable burning through CNO hydrogenshell flashes. However, for Z = 0.001 and Z = 0.0002 we note a



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Fig. 11: ∆tproto for the basic models (purple circles), for the models with diffusion only (orange stars), and for the models withdiffusion+rotation (blue squares) as a function of proto-WD mass for Z = 0.02 (top panels), Z = 0.01 (second panels), Z = 0.001(third panels) and Z = 0.0002 (bottom panels). The left-hand panels show a linear scale in ∆tproto while the right-hand panels showa log scale. Open symbols denote models that experience hydrogen shell flashes, whereas filled symbols denote models that avoidflashes. The grey shaded regions indicate temporary Roche-lobe detachment.

slight increase in ∆tproto around the lowest threshold for flashesfor all models where diffusion is included.

The maximum value of Teff reached during the proto-WDphase, however, is very sensitive to both the time and the spatialresolution with which the stellar structure is computed. With thisin mind, and taking into account that ∆tproto is relatively smallaround the lowest threshold for flashes at these low metallicitiesthe results shown in Fig. 11 do not present evidence for a di-chotomy in ∆tproto that is due to hydrogen flashes. However, forthe long-term evolution on the WD cooling track the situation isdifferent, as we discuss below.

4.5. Dichotomy on ELM WD cooling tracks

The hydrogen envelope mass is an important parameter that de-termines the long-term cooling timescale for WDs. Followingthe work of Istrate et al. (2014b), we consider the beginning ofthe cooling track as the moment at which the proto-WD reachesits maximum value of Teff . Figure 12 shows the remaining hydro-gen envelope mass when the proto-WD reaches the maximumTeff (MH,Teff,max ) and finally settles on the cooling track, as a func-tion of the mass of the proto-WD. Again, the large scatter is re-lated to the number of flashes (between 0−7) that the proto-WDexperiences. At all metallicities we note a jump in MH,Teff,max thatoccurs at the lowest threshold for flashes. At low metallicities,the effect is more pronounced in models with diffusion and diffu-sion+rotation, whereas at Z=0.02 the discontinuity is only seenin the basic models (all the systems for which element diffusion



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Fig. 12: Hydrogen envelope mass at the beginning of the coolingtrack (maximum Teff) for the basic models (purple circles), forthe models with diffusion only (orange stars) and for the modelswith diffusion+rotation (blue squares) as a function of proto-WDmass. See Fig. 11 for further explanations.

is considered experience hydrogen flashes). When element dif-fusion is included, the hydrogen envelope mass at the beginningof the cooling track is typically twice as small as when elementdiffusion is neglected (basic models), except for models at lowmetallicities and masses below the flash threshold. This signifi-cantly affects the cooling times of these objects.

For example, consider an ELM WD with a mass of ∼0.20 M. Figure 12 shows that flashes at high metallicities(Z = 0.02 and Z = 0.01) cause the amount of remaining hydro-gen envelope mass at T max

eff(MH,Teff,max ∼ 1.0× 10−3 M) to be up

to five times smaller than for the cases with lower metallicities(Z = 0.001 and Z = 0.0002) where flashes do not develop and athick ∼5.0×10−3 M residual hydrogen envelope remains at the

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Fig. 13: Timescale tcool,L−2 (see text) for the basic models (purplecircles), for the models with diffusion only (orange stars), and forthe models with diffusion+rotation (blue squares) as a functionof proto-WD mass. See Fig. 11 for further explanations.

onset of the cooling track. Hence, it is clear that we do see a di-chotomy in the long-term cooling ages of ELM WDs, such thatthose proto-WDs that experience flashes will have thin hydro-gen envelopes and therefore shorter cooling timescales, whereasproto-WDs that avoid flashes will have relatively thick hydrogenenvelopes and cool on a much longer timescale. It is importantto stress that the threshold mass at which this transition occurs isdependent on metallicity.

We now analyse the dichotomy in long-term cooling in moredetail. Figure 13 shows the time from the end of the LMXBmass transfer until the WD luminosity reaches log(L/L) =−2.0, tcool,L−2 (including ∆tproto). Some of our computed mod-els are not plotted because they reached an age of 14 Gyr beforelog(L/L) = −2.0. For the basic models, independent of metal-



3.43.63.84.04.24.44.64.8

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Fig. 14: Different cooling curves illustrating the differences be-tween basic models (without diffusion and rotation) and mod-els that include element diffusion and rotation. All models havea metallicity of Z = 0.001. The value of MWD/M is shownfor each track, and the difference in masses can partly explainthe different cooling rates. However, more important in this re-spect is the occurrence of hydrogen shell flashes (blue and greencurves), which accelerates the cooling and creates a dichotomyin WD cooling ages (see text).

licity, there is a small difference in cooling times between theWDs that experience flashes and those for which the hydrogenburning in the shell is stable. However, for the WDs computedwith diffusion and diffusion+rotation, the difference in coolingtimes between systems with and without flashes can be as largeas 3 Gyr, see Fig.14. We conclude that when element diffusionis included, the occurrence of hydrogen shell flashes does in-deed produce a (metallicity-dependent) dichotomy in the coolingtimes of helium WDs.

Only when element diffusion is neglected in the modellingthere is no or only a very small difference between the WDsthat experience unstable hydrogen burning compared to thosefor which the residual hydrogen burning is stable, independent ofmetallicity. This can explain the findings of Istrate et al. (2014b),who evolved their stellar models without element diffusion andthus questioned the dichotomy idea.

5. Discussion

5.1. Rotational mixing: source of surface metals?

In the past few years, metals, especially calcium, were detectedin the spectra of ELM WDs with a surface gravity lower than∼5.9. Metals sink below the atmosphere on a timescale muchshorter than the evolutionary timescale of the proto-WD, whichmeans that another process is required to either counteract thegravitational settling or replenish the depleted metals. There areseveral possible processes that can be responsible for the ob-served surface composition of ELM WDs. For a detailed dis-cussion, we refer to Gianninas et al. (2014a) and Hermes et al.(2014b). For higher mass (carbon-oxygen) WDs, the presenceof metals in their atmosphere is explained by accretion fromcircumstellar debris discs formed by tidal disruption of plane-tary bodies (e.g Debes & Sigurdsson 2002; Jura et al. 2007),which are detectable through excess flux in the IR (e.g Farihiet al. 2009; Kilic et al. 2006). This scenario seems unlikely forELM WDs given that their compact orbits make the existence ofa debris disk dynamically difficult to explain.

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H)

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Fig. 15: Evolution of log (Ca/H) at the stellar surface (top panel)and log g (bottom panel) for a 0.185 M proto-WD computedwith diffusion only (orange) and diffusion+rotation (blue) forZ=0.02. The starting point (t = 0) is defined at the moment ofRoche-lobe detachment.

Kaplan et al. (2013) suggested that the observed metals arebrought to the surface by the pulse-driven convection developedduring a hydrogen shell flash. However, shortly after the convec-tion zone vanishes, the metals will sink below the stellar surfaceas a result of gravitational settling.A mechanism that can counteract diffusion would be radiativelevitation. However, as Hermes et al. (2014a) showed, radiativelevitation alone cannot explain the observed abundances, espe-cially in the case of calcium. They suggested that in additionto radiative levitation, another support mechanism such as rota-tional mixing is likely required to explain the observed pattern inthe metal abundances of ELM WDs. Here, we discuss the effectof rotational mixing in determining the surface composition ofELM WDs.

Figure 15 shows the evolution of log (Ca/H) at the stellar sur-face from the beginning of the proto-WD phase (Roche-lobe de-tachment) to several hundred Myr onto the cooling track. In themodel that only includes diffusion calcium sinks much faster be-neath the surface than the proto-WD evolutionary timescale afterthe mass transfer ends. It is brought back to the surface throughto the pulse-driven convection zone that is developed by the oc-currence of a hydrogen shell flash, only to quickly sink againas a result of the gravitational settling. In the diffusion+rotationmodel, rotational mixing at the surface acts against the gravita-tional settling (see Sect. 4.1.1). As the proto-WD advances to-wards higher surface gravity, rotational mixing becomes less ef-ficient than gravitational settling. During the proto-WD phase,the star may experience several episodes of radial expansion fol-lowed by contraction and may also develop zones of convec-tion through the hydrogen flashes. This interplay between con-vection, expansion (low surface gravity), contraction (high sur-face gravity), and rotational mixing explains the pattern shownin Fig. 15. When the proto-WD enters the cooling track, the sur-face gravity steadily increases and gravitational settling finallyovercomes the mixing that is due to rotation. As a long-term re-



5101520253035

Teff (103 K)

56

78

log

g(c

ms−

2)

diffusion

−9.0

−8.5

−8.0

−7.5

−7.0

−6.5

−6.0

−5.5

−5.0

log

(Ca

/H

)

(a)

5101520253035

Teff (103 K)

56

78

log

g(c

ms−

2)

diffusion+rotation

−9.0

−8.5

−8.0

−7.5

−7.0

−6.5

−6.0

−5.5

−5.0

log

(Ca

/H

)

(b)

Fig. 16: Surface gravity versus effective temperature for the models with diffusion only (a), and diffusion+rotation (b), both calcu-lated for a metallicity of Z = 0.02. Each evolutionary track is plotted as a dot at 0.5 Myr intervals and colour-coded according tothe value of log (Ca/H) at the stellar surface. The over-plotted squares are the observed (proto)WDs with measured Ca abundancestaken from Gianninas et al. (2014a).

0 1 2 3 4 5 6 7 8 9 10 11 12

MH (10−3 M)

−4

−2

0

2

4

log

(Lcno/L

)

0.181 M, this work0.182 M, Althaus

1.2 1.3 1.4 1.5 1.6 −2

02

4

(a)

0 1 2 3 4 5 6 7

MH (10−3 M)

−4

−2

0

2

4

6lo

g(L

cno/L

)


1.4 1.5 1.6 1.7 1.8 1.9 −2

02

4

(b)

Fig. 17: Comparison of evolutionary tracks of Althaus et al. (2013) and this work (including diffusion only) for a ∼0.18 M WD (a)and ∼ 0.27 M WD (b). The post-RLO evolution of luminosity, given by CNO burning, is plotted as a function of the hydrogenenvelope mass. Time goes from right to left. The inset shows an artefact of hydrogen production during the shell flashes in theAlthaus et al. (2013) models, see text.

sult, the metals will sink below the surface, leaving behind a purehydrogen envelope.

We plott in Fig. 16 all the models with Z = 0.02 computedwith diffusion only (left panel) and diffusion+rotation (rightpanel). The points are spaced at intervals of 0.5 Myr and colour-coded according to the value of log (Ca/H). Over-plotted are thedata points from Gianninas et al. (2014a). The left panel clearlyshows that the flash scenario discussed above cannot explain theobservations. On the other hand, when rotational mixing is in-cluded, we can qualitatively explain the presence of calcium inthe spectra of proto-WDs as a natural result of their evolution.We recall that the observational data most likely belong to popu-lations with different metallicities, while in Fig. 16 we only plotour models with Z = 0.02. The lack of observations of proto-WDs at high Teff arises because the detection limit of Ca linesdepends on the effective temperature (see Fig. 9 in Gianninaset al. (2014a)).

5.2. Comparison with previous work

As discussed in Gianninas et al. (2015) and Bours et al. (2015),the models of Istrate et al. (2014b) (from here on I14) and Al-thaus et al. (2013)(A13) show a relatively large discrepancy intheir cooling ages. Although the initial binary parameters, andto some extent the metallicities, are different in the two sets ofmodels, the main difference is that the models of A13 includeelement diffusion, which has an important role in reducing thehydrogen envelope mass through flashes (cf. Sect. 4.5), and thusconsequently leads to accelerated cooling and therefore youngercooling ages than in I14 models.

Here, we compare our new models including element diffu-sion (but without rotational mixing to enable comparison) (I16)with the A13 models. We also adopted the same initial binaryparameters, namely a 1.0 M donor star and a 1.4 M neutronstar accretor, and also applied the same metallicity of Z = 0.01.A13 found that the lower mass limit for which hydrogen shell



51015202530

Teff (103 K)

4

5

6

7

8

log

g(c

ms−

2)

1 Gyr4 Gyr8 Gyr 0.181 M, this work

0.182 M, Althaus

(a)

01020304050607080

Teff (103 K)

1

2

3

4

5

6

7

8

log

g(c

ms−

2)

1 Gyr4 Gyr8 Gyr


(b)

Fig. 18: Evolutionary tracks in the (Teff , log g)–diagram for the same models as in Fig. 17. The circles, stars, and squares indicatecooling ages of 1, 4, and 8 Gyr, respectively. For the 0.18 M WD, the 1 Gyr symbols for the two models almost coincide. Moreover,we note that the models from this work are only calculated to an age of 14 Gyr, which prevents cooling ages exceeding 3 Gyr forthe 0.18 M model and 6 Gyr for the 0.27 M model.

flashes occur is somewhere in the interval 0.176 − 0.182 M(i.e. between the last model with stable shell burning and thefirst model that experiences flashes), while we find a lower masslimit of 0.165 − 0.169 M.

Figure 17 compares the I16 models with those of A13 byshowing the evolution of luminosity produced by CNO burningas a function of hydrogen envelope mass for a ∼ 0.18 M anda ∼ 0.27 M (proto) helium WD. One important difference isthe hydrogen envelope mass left at Roche-lobe detachment.For the 0.18 M WD, the model of I16 initially has a moremassive hydrogen envelope, while for the 0.27 M WD it isthe opposite. We mention again that in our models diffusionacts from the ZAMS, in contrast with the A13 models, wherediffusion is turned on during the proto-WD evolution. Figure 17also shows that the A13 models contain a numerical artefact bywhich hydrogen is created during CNO burning (see inset).

In Fig. 18 we present a (Teff , log g)–diagram and compare theevolutionary tracks from Fig. 17. The main difference are addi-tional mass-transfer episodes in the I16 models as a result of afew vigorous flashes. This effect will in the end leave a slightlylower WD mass at the beginning of the cooling track. These dif-ferences in proto-WD evolution, combined with the artificial cre-ation of hydrogen in the A13 models, result in slight differenceson the cooling track. However, the difference between the A13models and the I16 models are significantly smaller than whencomparing the A13 models with the I14 models (Gianninas et al.2015; Bours et al. 2015).

5.3. Relation of mass to orbital period in WDs

When low-mass stars (< 2.3 M) reach the red-giant branch, theradius of the star mainly depends on the mass of the degeneratehelium core and is almost entirely independent of the mass of theenvelope (Refsdal & Weigert 1971; Webbink et al. 1983). For theformation of binary MSPs, this relation proves to be very impor-tant because it provides a correlation between the mass of thenewly formed WD and Porb following the mass transfer episode(Savonije 1987; Joss et al. 1987; Rappaport et al. 1995; Tauris &Savonije 1999; Nelson et al. 2004; De Vito & Benvenuto 2010;

Lin et al. 2011; Shao & Li 2012; Jia & Li 2014; Istrate et al.2014a).

Figure 19 shows the (MWD, Porb)-relation for all the modelscomputed in this work. Our results are in fine agreement withTauris & Savonije (1999) for systems with Porb > 1 − 2 days.For close-orbit systems with Porb < 1 day our results agree wellwith Lin et al. (2011) and Istrate et al. (2014a). We note a slightdiscontinuity in the relation, which is dependent on metallicityas discussed in Sect. 4.3. This weak break in the (MWD, Porb)-relation was previously reported by other authors (e.g. Nelsonet al. 2004; Jia & Li 2014).

6. Conclusions

We computed a grid of models for ELM WDs with differentmetallicities. For each metallicity, we computed three types ofmodels with different physics included: i) basic models (with noelement diffusion nor rotation), ii) diffusion (including elementdiffusion), and iii) diffusion+rotation. For the first time, we tookinto account the combined effects of rotational mixing and el-ement diffusion in the evolution of WD progenitors and duringthe proto-WD phase and WD cooling. The main results obtainedare summarized as follows:

(i) We confirm that element diffusion plays a significant rolein the evolution of proto-WDs that experience hydrogenshell flashes. We also confirm that the unstable burning istriggered by the diffusive hydrogen tail reaching the hotdeep layers inside the star (Althaus et al. 2001a). The for-mation of the hydrogen tail is a cyclic process and dependson the available hydrogen in the envelope. Consequently,the number of flashes experienced by a proto-WD of agiven mass is increased, leading to reduced hydrogen en-velope mass and subsequent accelerated cooling, comparedwith the models without element diffusion.

(ii) Rotational mixing counteracts the effect of gravitationalsettling in the surface layers of young bloated ELM proto-WDs, but its efficiency is reduced towards the end of theproto-WD phase, when the star contracts and its surfacegravity increases. As a consequence, our new evolution-ary models including rotational mixing predict that ELM



100

101

102

103

Porb

(day

s)

Z=0.02


temporary detachment

TS99-Pop ILin+11

100

101

102

103

Porb

(day

s)

Z=0.01



TS99-Pop ILin+11

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101

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Porb

(day

s)

Z=0.001



TS99-Pop ITS99-Pop II

0.15 0.20 0.25 0.30 0.35 0.40 0.45

proto-WD mass (M)

100

101

102

103

Porb

(day

s)

Z=0.0002


temporarydetachment

TS99-Pop ITS99-Pop II

Fig. 19: (MWD, Porb)-relation for Z = 0.02 (top panel), Z = 0.01(second panel), Z = 0.001 (third panel), and Z = 0.0002 (bot-tom panel). The grey shaded area denotes the LMXB systemsthat undergo a temporary detachment (see text). The purple cir-cles represent the basic models, the orange stars the modelswith diffusion only and the blue squares the models with diffu-sion+rotation. Over-plotted are the theoretical relations by Tau-ris & Savonije (1999) and Lin et al. (2011) for the respectivemetallicity.

proto-WDs have mixed H/He envelopes during a signif-icant part of their evolution before settling on the cool-ing track, in accordance with recent observational evi-dence from pulsations in ELM proto-WDs (Gianninas et al.2016). Except for this, the general properties, such as thenumber of flashes, are not strongly influenced by the pres-ence of rotational mixing.

(iii) Although the hydrogen envelope left after detachment fromthe LMXB phase is a small fraction of the total WD mass,

it has a very high angular momentum content compared tothe core. Because the proto-WD contracts while this hy-drogen is burnt and angular momentum mixes inwards fol-lowing each flash, the resulting WD on the cooling trackis spun up significantly to a rotation period well below theorbital period.

(iv) The hydrogen envelope mass in newborn proto-WDs is in-fluenced by the evolutionary stage of the donor star at themoment when LMXB mass transfer is initiated. In particu-lar, we found that LMXB donor stars that experience a tem-porary contraction will produce proto-WDs with a signifi-cantly reduced hydrogen envelope mass at the moment ofthe final Roche-lobe detachment. In general, the shorter theorbital period at the onset of the LMXB phase, the lowerthe mass of the proto-WD and the higher the final envelopemass. The hydrogen envelope mass is also metallicity de-pendent, such that (for the same proto-WD mass) the lowerthe metallicity, the higher the envelope mass.

(v) In general, our resulting mass range for the occurrenceof flashes is similar to those found in the literature. ForZ = 0.02, all our models with diffusion experience flashes,and for Z = 0.01 we obtain a lower limit of ∼ 0.16 M,compared to ∼0.18 M found by Althaus et al. (2013).

(vi) We identified two timescales relevant for understandingthe evolution of (proto) WDs. The evolutionary timescaleof the contraction of proto-WDs, ∆tproto, can reach up to2.5 Gyr for the lowest proto-WD masses, while for thehigher WD masses it is < 100 Myr. When element dif-fusion is included, this timescale is slightly increased bymore numerous flashes than for the case when element dif-fusion is neglected. As concluded in Istrate et al. (2014b),we did not find a dichotomy in the ∆tproto distribution withrespect to the occurrence of flashes, but rather a smoothtransition. However, we note a small increase in ∆tprotoclose to the flash limit for Z = 0.001 and Z = 0.002.The cooling timescale tcool,L−2 describes the evolution of theWD from LMXB detachment until the the WD has evolvedwell down the cooling track and reaches log (L/L) = −2.This timescale is mostly determined by the hydrogen en-velope mass that remains after the proto-WD phase. In thebasic models (without element diffusion or rotation), inde-pendent of metallicity, there is a very small difference be-tween the models that experience flashes and those that donot. However, we confirm, as first stated by Althaus et al.(2001a), that the situation is different when element diffu-sion is included, which leads to a dichotomy in the coolingtimescale of ELM WDs.

(vii) We investigated whether the observed metal features inELM proto-WDs might be linked to the internal evolutionof these objects. In particular, we analysed the evolution ofthe surface abundance of calcium and concluded that ro-tational mixing is a key component for producing the ob-served pattern in the (log g,Teff)–diagram.

The systematic investigation presented here for the effects ofthermal and chemical diffusion, gravitational settling, and ro-tational mixing on a wide range of LMXBs with different ini-tial masses, orbital periods, and metallicity, leads us to concludethat theoretical models must include these aspects when they arecompared to observational data of ELM WDs. Hence, with theseimproved models at hand, the implications are better constraintson the true ages and masses of these WDs and therefore also onthe ages and masses of their companion stars, such as millisec-ond radio pulsars. Moreover, the grid of models we presented



might be further used for astroseismology calculations to deter-mine the pulsational behaviour of ELM proto-WDs and ELMWDs.

Acknowledgements

We thank the anonymous referee for helpful advice. A.G.I.thanks Alejandra Romero and Elvijs Matrozis for very usefuldiscussions and BWin Technologies Ltd. for the computing coreson which the simulations in this work were performed.

ReferencesAlthaus, L. G. & Benvenuto, O. G. 2000, MNRAS, 317, 952Althaus, L. G., Miller Bertolami, M. M., & Córsico, A. H. 2013, A&A, 557, A19Althaus, L. G., Panei, J. A., Romero, A. D., et al. 2009, A&A, 502, 207Althaus, L. G., Serenelli, A. M., & Benvenuto, O. G. 2001a, MNRAS, 323, 471Althaus, L. G., Serenelli, A. M., & Benvenuto, O. G. 2001b, ApJ, 554, 1110Althaus, L. G., Serenelli, A. M., & Benvenuto, O. G. 2001c, MNRAS, 324, 617Amaro-Seoane, P., Aoudia, S., Babak, S., et al. 2012, Classical and Quantum

Gravity, 29, 124016Andrews, J. J., Price-Whelan, A. M., & Agüeros, M. A. 2014, ApJ, 797, L32Antoniadis, J., van Kerkwijk, M. H., Koester, D., et al. 2012, MNRAS, 423, 3316Bassa, C. G., Antoniadis, J., Camilo, F., et al. 2016, MNRAS, 455, 3806Bildsten, L., Shen, K. J., Weinberg, N. N., & Nelemans, G. 2007, ApJ, 662, L95Bours, M. C. P., Marsh, T. R., Gänsicke, B. T., et al. 2015, MNRAS, 450, 3966Brown, W. R., Gianninas, A., Kilic, M., Kenyon, S. J., & Allende Prieto, C. 2016,

ApJ, 818, 155Brown, W. R., Kilic, M., Allende Prieto, C., Gianninas, A., & Kenyon, S. J. 2013,

ApJ, 769, 66Brown, W. R., Kilic, M., Allende Prieto, C., & Kenyon, S. J. 2010, ApJ, 723,

1072Brown, W. R., Kilic, M., Allende Prieto, C., & Kenyon, S. J. 2012, ApJ, 744, 142Cadelano, M., Pallanca, C., Ferraro, F. R., et al. 2015, ApJ, 812, 63Camilo, F., Thorsett, S. E., & Kulkarni, S. R. 1994, ApJ, 421, L15Cassisi, S., Potekhin, A. Y., Pietrinferni, A., Catelan, M., & Salaris, M. 2007,

ApJ, 661, 1094Christensen-Dalsgaard, J. 2015, MNRAS, 453, 666Clayton, G. C. 2013, in Astronomical Society of the Pacific Conference Series,

Vol. 469, 18th European White Dwarf Workshop., 133Córsico, A. H. & Althaus, L. G. 2014a, A&A, 569, A106Córsico, A. H. & Althaus, L. G. 2014b, ApJ, 793, L17Córsico, A. H., Althaus, L. G., Serenelli, A. M., et al. 2016, A&A, 588, A74Corti, M. A., Kanaan, A., Córsico, A. H., et al. 2016, A&A, 587, L5De Vito, M. A. & Benvenuto, O. G. 2010, MNRAS, 401, 2552Debes, J. H. & Sigurdsson, S. 2002, ApJ, 572, 556Detmers, R. G., Langer, N., Podsiadlowski, P., & Izzard, R. G. 2008, A&A, 484,

831Driebe, T., Schoenberner, D., Bloecker, T., & Herwig, F. 1998, A&A, 339, 123Farihi, J., Jura, M., & Zuckerman, B. 2009, ApJ, 694, 805Ferguson, J. W., Alexander, D. R., Allard, F., et al. 2005, ApJ, 623, 585Foley, R. J. 2015, MNRAS, 452, 2463Fontaine, G. & Michaud, G. 1979, ApJ, 231, 826Gautschy, A. 2013, ArXiv e-printsGianninas, A., Curd, B., Fontaine, G., Brown, W. R., & Kilic, M. 2016, ArXiv

e-printsGianninas, A., Dufour, P., Kilic, M., et al. 2014a, ApJ, 794, 35Gianninas, A., Hermes, J. J., Brown, W. R., et al. 2014b, ApJ, 781, 104Gianninas, A., Kilic, M., Brown, W. R., Canton, P., & Kenyon, S. J. 2015, ApJ,

812, 167Grevesse, N. & Sauval, A. J. 1998, Space Sci. Rev., 85, 161Heber, U. 2016, ArXiv e-printsHeger, A., Langer, N., & Woosley, S. E. 2000, ApJ, 528, 368Heger, A., Woosley, S. E., & Spruit, H. C. 2005, ApJ, 626, 350Henyey, L., Vardya, M. S., & Bodenheimer, P. 1965, ApJ, 142, 841Hermes, J. J., Brown, W. R., Kilic, M., et al. 2014a, ApJ, 792, 39Hermes, J. J., Gänsicke, B. T., Koester, D., et al. 2014b, MNRAS, 444, 1674Hermes, J. J., Kilic, M., Brown, W. R., et al. 2012a, ApJ, 757, L21Hermes, J. J., Montgomery, M. H., Gianninas, A., et al. 2013a, MNRAS, 436,

3573Hermes, J. J., Montgomery, M. H., Winget, D. E., et al. 2013b, ApJ, 765, 102Hermes, J. J., Montgomery, M. H., Winget, D. E., et al. 2012b, ApJ, 750, L28Hu, H., Tout, C. A., Glebbeek, E., & Dupret, M.-A. 2011, MNRAS, 418, 195Hurley, J. R., Tout, C. A., & Pols, O. R. 2002, MNRAS, 329, 897Iben, Jr., I. & MacDonald, J. 1985, ApJ, 296, 540Iben, Jr., I. & Tutukov, A. V. 1984, ApJS, 54, 335

Iglesias, C. A. & Rogers, F. J. 1993, ApJ, 412, 752Iglesias, C. A. & Rogers, F. J. 1996, ApJ, 464, 943Istrate, A. G., Tauris, T. M., & Langer, N. 2014a, A&A, 571, A45Istrate, A. G., Tauris, T. M., Langer, N., & Antoniadis, J. 2014b, A&A, 571, L3Itoh, N., Kohyama, Y., & Takeuchi, H. 1987, ApJ, 317, 733Jeffery, C. S. & Saio, H. 2013, MNRAS, 435, 885Jia, K. & Li, X.-D. 2014, ApJ, 791, 127Joss, P. C., Rappaport, S., & Lewis, W. 1987, ApJ, 319, 180Jura, M., Farihi, J., & Zuckerman, B. 2007, ApJ, 663, 1285Kaplan, D. L., Bhalerao, V. B., van Kerkwijk, M. H., et al. 2013, ApJ, 765, 158Kaplan, D. L., Stovall, K., Ransom, S. M., et al. 2012, ApJ, 753, 174Kepler, S. O., Pelisoli, I., Koester, D., et al. 2016, MNRAS, 455, 3413Kilic, M., Brown, W. R., Allende Prieto, C., et al. 2011, ApJ, 727, 3Kilic, M., Brown, W. R., Allende Prieto, C., et al. 2012, ApJ, 751, 141Kilic, M., Brown, W. R., & Hermes, J. J. 2013, in Astronomical Society of the

Pacific Conference Series, Vol. 467, 9th LISA Symposium, ed. G. Auger,P. Binétruy, & E. Plagnol, 47

Kilic, M., Hermes, J. J., Gianninas, A., & Brown, W. R. 2015, MNRAS, 446,L26

Kilic, M., Hermes, J. J., Gianninas, A., et al. 2014, MNRAS, 438, L26Kilic, M., Stanek, K. Z., & Pinsonneault, M. H. 2007, ApJ, 671, 761Kilic, M., von Hippel, T., Leggett, S. K., & Winget, D. E. 2006, ApJ, 646, 474Koester, D., Voss, B., Napiwotzki, R., et al. 2009, A&A, 505, 441Latour, M., Heber, U., Irrgang, A., et al. 2016, A&A, 585, A115Lin, J., Rappaport, S., Podsiadlowski, P., et al. 2011, ApJ, 732, 70Lorimer, D. R., Lyne, A. G., Festin, L., & Nicastro, L. 1995, Nature, 376, 393Marsh, T. R., Dhillon, V. S., & Duck, S. R. 1995, MNRAS, 275, 828Maxted, P. F. L., Anderson, D. R., Burleigh, M. R., et al. 2011, MNRAS, 418,

1156Maxted, P. F. L., Bloemen, S., Heber, U., et al. 2014a, MNRAS, 437, 1681Maxted, P. F. L., Serenelli, A. M., Marsh, T. R., et al. 2014b, MNRAS, 444, 208Maxted, P. F. L., Serenelli, A. M., Miglio, A., et al. 2013, Nature, 498, 463Nandez, J. L. A., Ivanova, N., & Lombardi, J. C. 2015, MNRAS, 450, L39Nelemans, G. & Tauris, T. M. 1998, A&A, 335, L85Nelson, L. A., Dubeau, E., & MacCannell, K. A. 2004, ApJ, 616, 1124Panei, J. A., Althaus, L. G., Chen, X., & Han, Z. 2007, MNRAS, 382, 779Paquette, C., Pelletier, C., Fontaine, G., & Michaud, G. 1986, ApJS, 61, 197Paxton, B., Bildsten, L., Dotter, A., et al. 2011, ApJS, 192, 3Paxton, B., Cantiello, M., Arras, P., et al. 2013, ApJS, 208, 4Paxton, B., Marchant, P., Schwab, J., et al. 2015, ApJS, 220, 15Perets, H. B., Gal-Yam, A., Mazzali, P. A., et al. 2010, Nature, 465, 322Rappaport, S., Podsiadlowski, P., Joss, P. C., Di Stefano, R., & Han, Z. 1995,

MNRAS, 273, 731Refsdal, S. & Weigert, A. 1971, A&A, 13, 367Rivera-Sandoval, L. E., van den Berg, M., Heinke, C. O., et al. 2015, MNRAS,

453, 2707Rohrmann, R. D., Althaus, L. G., García-Berro, E., Córsico, A. H., & Miller

Bertolami, M. M. 2012, A&A, 546, A119Sarna, M. J., Ergma, E., & Gerškevitš-Antipova, J. 2000, MNRAS, 316, 84Savonije, G. J. 1987, Nature, 325, 416Serenelli, A. M., Althaus, L. G., Rohrmann, R. D., & Benvenuto, O. G. 2001,

MNRAS, 325, 607Serenelli, A. M., Althaus, L. G., Rohrmann, R. D., & Benvenuto, O. G. 2002,

MNRAS, 337, 1091Shao, Y. & Li, X.-D. 2012, ApJ, 756, 85Solheim, J.-E. 2010, PASP, 122, 1133Spruit, H. C. 2002, A&A, 381, 923Tauris, T. M. 2012, Science, 335, 561Tauris, T. M., Langer, N., & Kramer, M. 2012, MNRAS, 425, 1601Tauris, T. M. & Savonije, G. J. 1999, A&A, 350, 928Thomas, H.-C. 1967, ZAp, 67, 420Thoul, A. A., Bahcall, J. N., & Loeb, A. 1994, ApJ, 421, 828Van Grootel, V., Fontaine, G., Brassard, P., & Dupret, M.-A. 2013, ApJ, 762, 57van Kerkwijk, M. H., Bassa, C. G., Jacoby, B. A., & Jonker, P. G. 2005, in As-

tronomical Society of the Pacific Conference Series, Vol. 328, Binary RadioPulsars, ed. F. A. Rasio & I. H. Stairs, 357

Vauclair, G., Vauclair, S., & Greenstein, J. L. 1979, A&A, 80, 79Webbink, R. F., Rappaport, S., & Savonije, G. J. 1983, ApJ, 270, 678



Appendix A: Properties of the computed models



Table A.1: Properties of selected models. The quantities given are (Cols. 1-7): the (proto) WD mass, the number of hydrogen shellflashes, the orbital period and the hydrogen envelope mass at Roche-lobe detachment, the hydrogen envelope mass at maximum Teff ,∆tproto, and the time interval (cooling timescale) from Roche-lobe detachment until the WD reaches a luminosity of log(L/L) = −2.

.

Z = 0.02, basic models (no element diffusion nor rotation)

Mass (M) # flashes Porb,det (d) MH,det (10−2 M) MH,Teff,max (10−3 M) ∆tproto (Myr) tcool,L−2 (Myr)0.171 0 0.451 0.827 3.02 1500 58200.180 0 0.767 0.867 3.01 1000 56300.180 0 0.779 0.869 3.01 993 56200.184 0 0.961 0.887 3.05 846 48500.191 0 1.38 0.952 3.01 672 48600.202 0 2.14 1.13 2.93 430 40600.205 0 2.34 1.19 2.91 392 39000.206 0 2.43 1.23 2.90 377 38500.208 0 2.53 1.26 2.88 364 37600.209 0 2.63 1.30 2.86 352 36900.210 0 2.73 1.34 2.83 342 36300.211 0 2.71 0.817 2.81 174 34300.213 3 2.94 0.808 1.72 274 25300.213 3 2.94 0.808 1.79 268 25700.214 3 3.05 0.804 1.68 261 24900.216 3 3.28 0.794 1.64 237 24100.216 3 3.40 0.790 1.59 221 23500.219 3 3.76 0.777 1.55 200 22800.233 2 6.89 0.692 1.50 97.1 20900.242 2 9.81 0.646 1.16 107 18400.250 2 12.8 0.614 0.905 153 15700.256 1 16.0 0.587 1.26 44.6 18100.261 1 19.0 0.566 1.22 46.1 18000.265 1 21.9 0.549 1.13 48.0 17200.268 1 24.5 0.534 1.05 50.1 16600.271 1 26.9 0.523 0.996 52.2 16100.274 1 29.3 0.513 0.945 54.3 15500.276 1 31.5 0.504 0.889 56.3 14900.278 1 33.7 0.496 0.868 58.2 14600.280 1 35.8 0.489 0.835 60.1 14100.282 1 37.8 0.482 0.805 61.9 13600.283 1 39.8 0.476 0.779 63.7 13200.285 1 41.7 0.470 0.755 65.4 12700.287 1 45.5 0.460 0.670 68.9 10700.305 1 75.6 0.404 0.451 105 4770.319 0 111 0.365 0.965 1.76 13100.343 0 202 0.307 0.775 0.838 11300.378 0 414 0.255 0.581 0.351 907



Table A.2: Properties of selected models. See Table A.1 for a description of the parameters.

Z = 0.02, diffusion

Mass (M) # flashes Porb,det (d) MH,det (10−2 M) MH,Teff,max (10−3 M) ∆tproto (Myr) tcool,L−2 (Myr)0.167 24 0.401 0.823 0.951 2230 26100.168 26 0.431 0.843 1.12 2130 26500.172 12 0.541 0.885 0.818 1940 21900.174 10 0.623 0.904 0.557 2050 20800.186 6 1.08 0.948 0.899 1160 15000.191 6 1.35 0.963 0.674 1020 11300.195 6 1.63 0.984 0.563 907 9450.202 5 2.14 1.05 0.689 636 7550.210 4 2.72 0.811 0.930 316 6380.215 4 3.40 0.775 0.806 264 4950.218 5 3.89 0.753 0.662 256 3920.230 3 6.45 0.672 0.753 134 3870.235 3 7.79 0.644 0.606 129 2990.239 4 9.16 0.620 0.540 121 2630.243 3 10.6 0.600 0.448 123 2340.246 4 12.0 0.582 0.390 128 2180.260 3 19.0 0.526 0.191 184 2050.277 3 32.6 0.467 0.365 54.9 2420.286 3 43.2 0.441 0.316 52.1 2460.298 2 60.0 0.409 0.246 50.7 2420.317 1 97.5 0.363 0.487 11.5 3280.333 2 157 0.331 0.416 9.96 3220.344 2 201 0.307 0.366 9.17 3080.393 2 532 0.242 0.211 7.03 298




Z = 0.02, diffusion+rotation

Mass (M) # flashes Porb,det (d) MH,det (10−2 M) MH,Teff,max (10−3 M) ∆tproto (Myr) tcool,L−2 (Myr)0.167 23 0.412 0.848 1.07 2340 27700.175 10 0.628 0.929 0.794 1910 21700.182 8 0.886 0.948 0.725 1430 16000.185 7 1.03 0.954 0.821 1220 15000.190 6 1.33 0.963 0.956 976 13500.198 6 1.92 0.992 0.981 666 10300.205 5 2.44 1.05 0.691 558 6840.212 4 3.03 0.775 0.801 309 5310.215 4 3.51 0.739 0.755 266 4600.217 5 3.76 0.723 0.659 265 3980.220 5 4.27 0.692 0.572 253 3390.221 4 4.53 0.678 0.557 234 3190.226 4 5.61 0.627 0.336 288 3030.232 3 7.01 0.572 0.719 119 3610.236 3 8.44 0.539 0.563 117 2720.241 3 9.88 0.517 0.476 116 2340.248 3 12.8 0.485 0.346 127 2040.261 3 20.0 0.438 0.578 51.1 2990.266 3 23.4 0.494 0.502 53.4 2750.278 3 33.6 0.463 0.365 54.8 2470.284 2 39.9 0.446 0.319 52.8 2430.287 2 43.8 0.438 0.299 52.7 2420.290 2 47.4 0.431 0.274 52.3 2380.293 2 52.6 0.421 0.273 51.8 2410.306 2 74.2 0.389 0.208 51.4 2410.317 1 97.6 0.363 0.491 11.6 3270.322 1 108 0.353 0.469 11.1 326





Mass (M) # flashes Porb,det (d) MH,det (10−2 M) MH,Teff,max (10−3 M) ∆tproto (Myr) tcool,L−2 (Myr)0.176 0 0.676 0.981 3.16 869 0.000.192 0 1.58 0.937 3.07 371 22000.205 0 2.76 0.944 3.08 277 11200.217 0 4.05 1.16 3.06 188 33100.222 4 4.56 1.32 1.77 281 24600.228 3 5.05 1.51 1.66 234 22900.230 3 5.19 0.804 1.65 125 21200.237 2 6.71 0.756 1.33 122 19200.243 2 8.26 0.721 1.60 70.3 20200.248 2 9.82 0.694 1.43 71.6 19300.256 1 12.7 0.654 1.16 89.2 17400.261 1 15.2 0.627 1.55 33.9 18900.269 1 19.7 0.593 1.32 34.7 17400.273 1 21.7 0.581 1.26 35.6 17200.284 1 30.5 0.539 1.03 41.2 15400.297 1 44.2 0.494 0.787 51.0 13200.305 1 55.5 0.467 0.669 59.6 11100.375 0 277 0.310 0.698 0.487 1010




Z = 0.01, diffusion

Mass (M) # flashes Porb,det (d) MH,det (10−2 M) MH,Teff,max (10−3 M) ∆tproto (Myr) tcool,L−2 (Myr)0.165 0 0.340 0.904 4.45 1580 0.000.170 7 0.458 0.946 0.807 2380 27500.171 5 0.501 0.950 0.967 2320 27800.175 5 0.651 1.02 0.980 1910 23500.181 7 0.937 1.05 0.589 1500 15900.203 6 2.97 0.927 0.804 487 7310.212 5 4.26 1.01 0.707 390 5720.223 5 5.53 1.30 0.448 405 4470.232 4 6.74 0.784 0.593 183 3390.238 4 8.33 0.745 0.411 201 2680.240 2 8.99 0.729 1.23 130 18100.243 3 9.98 0.712 0.808 94.8 4150.248 3 11.6 0.685 0.654 95.9 3350.252 3 13.2 0.664 0.561 97.2 2930.255 4 14.7 0.646 0.475 102 2610.260 3 17.4 0.617 0.371 114 2320.265 3 19.9 0.596 0.299 127 2190.271 3 24.2 0.566 0.658 47.1 3420.274 2 26.2 0.555 0.572 47.6 3110.279 3 29.9 0.536 0.534 48.5 3010.283 2 33.2 0.522 0.474 49.2 2830.286 2 36.4 0.510 0.425 49.6 2680.291 2 42.0 0.491 0.377 50.3 260





Mass (M) # flashes Porb,det (d) MH,det (10−2 M) MH,Teff,max (10−3 M) ∆tproto (Myr) tcool,L−2 (Myr)0.182 7 1.01 1.04 0.873 1390 17400.183 6 1.09 1.04 0.898 1270 16200.187 7 1.34 1.04 0.946 1070 14600.192 7 1.72 1.02 0.957 823 12100.195 7 1.92 1.02 0.790 777 10200.197 6 2.14 0.983 0.995 615 10200.206 5 3.32 0.958 0.689 491 6520.216 5 4.59 1.09 0.824 339 6180.226 4 5.79 1.41 0.870 264 5960.234 4 7.05 0.775 0.543 187 3170.239 4 8.58 0.740 0.399 199 2620.244 3 10.2 0.709 0.723 102 3780.248 3 11.7 0.684 0.660 95.1 3420.252 4 13.2 0.663 0.565 98.9 2930.258 3 16.0 0.631 0.401 111 2400.262 3 18.5 0.608 0.326 121 2220.265 3 19.9 0.596 0.299 127 2190.275 2 26.8 0.551 0.579 48.1 3140.279 2 30.4 0.533 0.501 48.9 2890.301 2 54.3 0.460 0.290 52.3 2510.324 1 90.6 0.401 0.556 12.3 368





Mass (M) # flashes Porb,det (d) MH,det (10−2 M) MH,Teff,max (10−3 M) ∆tproto (Myr) tcool,L−2 (Myr)0.182 0 0.471 1.48 3.74 1100 26400.190 0 0.674 1.55 3.71 816 28600.207 0 1.45 1.58 3.64 387 35500.211 0 1.75 1.56 3.62 314 34100.228 0 3.49 1.44 3.44 142 27600.234 0 4.62 1.37 3.32 102 26100.238 0 5.38 1.32 3.25 85.5 24700.246 0 7.22 1.23 3.10 60.2 23100.252 3 8.90 1.17 1.85 88.9 17600.256 4 10.4 1.10 1.87 72.2 17200.260 4 11.8 1.05 1.63 73.2 16300.263 3 13.0 1.02 1.79 60.6 16900.266 3 14.0 0.999 1.97 61.7 17800.274 3 17.5 1.01 1.48 61.8 15400.282 3 20.0 1.16 1.27 65.5 14300.289 2 22.0 1.42 1.55 43.1 15100.302 2 26.9 0.861 1.11 45.5 13100.309 1 31.9 0.815 1.56 15.6 14500.328 1 49.1 0.708 1.16 16.6 12700.340 1 63.7 0.648 0.950 18.9 11500.350 1 76.9 0.607 0.810 21.4 10900.423 0 258 0.385 0.781 0.406 875




Z = 0.001, diffusion

Mass (M) # flashes Porb,det (d) MH,det (10−2 M) MH,Teff,max (10−3 M) ∆tproto (Myr) tcool,L−2 (Myr)0.180 0 0.466 1.50 5.00 1280 0.000.183 5 0.582 1.58 4.86 1160 0.000.190 0 0.800 1.64 4.63 886 59000.196 0 1.05 1.67 4.44 692 58300.217 1 2.65 1.60 3.87 261 39200.230 4 4.59 1.46 1.21 289 8050.239 6 6.61 1.35 0.840 227 5890.245 5 8.54 1.27 0.958 155 5800.250 2 10.3 1.20 0.674 168 4510.254 2 11.9 1.15 0.651 153 4290.264 2 16.7 1.01 0.668 122 4120.268 1 18.5 0.985 0.518 149 3600.271 1 19.9 0.986 0.462 153 3410.280 3 23.4 1.11 0.756 77.6 4380.314 1 41.4 0.785 0.339 76.9 2820.322 3 49.7 0.734 0.676 28.1 452



Mass (M) # flashes Porb,det (d) MH,det (10−2 M) MH,Teff,max (10−3 M) ∆tproto (Myr) tcool,L−2 (Myr)0.179 0 0.477 1.54 5.30 1310 0.000.183 0 0.572 1.59 5.18 1140 0.000.190 0 0.783 1.65 4.86 891 23000.196 0 1.03 1.68 4.62 702 22300.216 0 2.59 1.59 4.02 270 41800.228 0 4.55 1.43 3.74 144 28000.237 4 6.58 1.32 1.18 233 0.000.244 1 8.49 1.23 0.646 246 4960.250 5 10.2 1.18 0.986 138 5820.254 5 11.7 1.16 0.803 132 4870.260 5 14.4 1.07 0.808 109 4790.275 1 21.1 1.03 0.443 144 3260.287 1 25.0 1.35 0.489 102 3320.296 2 27.3 0.910 0.725 50.4 4350.303 1 32.3 0.858 0.558 58.0 3510.314 1 41.5 0.785 0.341 78.9 2840.322 1 49.7 0.734 0.644 28.3 428





Mass (M) # flashes Porb,det (d) MH,det (10−2 M) MH,Teff,max (10−3 M) ∆tproto (Myr) tcool,L−2 (Myr)0.214 0 0.900 2.10 3.78 533 35500.227 0 1.61 2.10 3.74 318 29800.247 0 3.86 1.90 3.58 131 23500.260 0 6.21 1.74 3.42 76.6 21100.268 0 8.30 1.63 3.28 54.9 19400.274 0 10.1 1.56 3.17 43.9 18700.278 0 11.6 1.51 3.10 37.4 18100.282 0 12.8 1.47 3.03 33.2 17800.284 1 13.9 1.43 2.44 41.0 17500.287 3 14.9 1.40 1.98 47.6 14800.289 4 15.8 1.38 1.84 47.0 14100.290 4 16.6 1.36 1.83 45.6 14100.292 4 17.3 1.34 1.73 44.7 13700.293 3 18.0 1.32 1.96 36.1 14200.295 3 18.7 1.31 1.82 37.0 13700.297 3 19.9 1.28 1.60 39.2 13100.302 4 23.1 1.20 1.50 36.9 12700.305 3 25.0 1.16 1.90 25.6 13800.308 4 26.8 1.12 1.61 31.8 13100.313 3 30.2 1.07 1.65 27.2 13000.317 2 33.4 1.04 1.85 20.0 13400.320 2 34.9 1.05 1.74 20.4 13000.322 2 36.4 1.06 1.65 21.0 12800.327 2 39.1 1.12 1.47 23.2 12100.329 2 40.4 1.18 1.38 25.0 11800.336 2 43.1 1.39 1.13 33.4 11000.340 1 44.6 1.55 1.68 16.0 12500.342 1 45.5 1.06 1.62 11.6 12200.356 1 57.5 0.901 1.32 10.7 11100.441 0 232 0.514 0.940 0.552 829




Z = 0.0002, diffusion

Mass (M) # flashes Porb,det (d) MH,det (10−2 M) MH,Teff,max (10−3 M) ∆tproto (Myr) tcool,L−2 (Myr)0.226 0 1.78 2.23 4.18 374 40600.234 0 2.54 2.17 4.01 265 36700.244 0 4.00 2.03 3.81 166 32400.246 0 4.40 2.00 3.77 149 31300.255 0 6.38 1.86 3.60 98.3 28300.263 0 8.56 1.74 3.43 69.8 26400.269 5 10.5 1.66 1.19 122 7350.273 0 12.1 1.60 3.22 46.4 24200.277 4 13.5 1.55 0.919 110 5690.282 4 15.9 1.48 0.944 89.1 5730.286 5 17.8 1.43 0.952 76.1 5740.289 5 19.4 1.39 0.678 100 4410.290 5 20.1 1.38 0.644 103 4200.291 4 20.8 1.36 1.03 59.3 6390.297 4 24.3 1.28 1.04 48.8 6640.300 4 26.3 1.24 0.899 53.4 5700.311 5 34.8 1.08 0.657 66.5 412



Mass (M) # flashes Porb,det (d) MH,det (10−2 M) MH,Teff,max (10−3 M) ∆tproto (Myr) tcool,L−2 (Myr)0.212 0 0.902 2.17 4.79 621 0.000.236 0 2.90 2.13 4.11 229 32900.250 0 5.32 1.92 3.77 121 31000.259 0 7.67 1.76 3.61 80.9 28300.266 0 9.76 1.65 3.51 61.8 0.000.270 0 11.5 1.57 3.46 51.0 25900.274 0 13.1 1.52 3.40 44.5 25600.277 3 14.4 1.47 1.08 115 6880.280 5 15.6 1.43 1.19 89.2 7790.282 5 16.6 1.40 1.10 89.1 6920.284 4 17.5 1.38 0.875 103 5420.286 5 18.4 1.36 0.892 95.9 5560.287 5 19.1 1.34 0.836 94.3 5190.288 5 19.9 1.32 0.893 84.0 5580.290 5 20.6 1.32 0.872 82.0 5380.291 5 21.2 1.31 0.897 75.2 5520.293 5 21.8 1.30 0.868 74.9 541


models of low-mass helium white dwarfs including gravitational … · 2016-06-17 · astronomy...

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